In a pinch you could take the largest garbage bag you have available, and cut open the side and bottom to create a single larger sheet. Then lay that don on the bed and top that with a few large towels.
I think the biggest problem I'd have with that is restless sleeping. You might be able to secure the assembly together with duct tape around the edges maybe?
Alternately, if you're more concerned with the mattress than the sheets, you can cut up a trash bag as above and slip it in between the bet sheet and the mattress. If you have a leak it'll wet the sheet but not the mattress. (as long as it's not a huge leak). This will also be very unlikely to shift if you move around, and may be a little cooler than a towel. It will be a little warmer than without the bag because you normally get a little cooling and ventilation through the mattress and the bag will block that. (I've done this technique before, it works pretty well) You could probably do this same technique with an actual bed pad (or puppy pad! those are cheap at pet stores and walmarts!) because the top sheet will stop the pad from moving around. You will still have a little laundry to do in the morning but it's a good tradeoff and protects the mattress.
The towel option will offer superior capacity though and intends to save you from needing to wash your sheets, though the towels will still need to be washed so you're not going to be able to avoid doing laundry one way or another if you leak.
Thinking more on this, you could combine the above ideas. Take off the bed sheet. Lay down the bag, and duct tape it to the mattress. (optional, lightly just to keep it in position as you set up) Lay down a few towels to cover the plastic. Then put the mattress sheet back on. You'd get the capacity of the towels, the mattress protection of the bag, and the comfort / cooler bed / lack of shifting materials of the bed sheet. Hmmm a how-to on this may make a good addition to my gallery.